ZURAB JANELIDZE'S BLOG

Below are the notes for the talk above, given at the Algebra, Geometry, Topology & Applications seminar. 1. Bird's-Eye View of Exactness Properties  One of the active areas of research in Categorical Algebra is the study of various properties of categories expressed using limits and colimits. Such properties are usually referred to as exactness properties . This terminology comes from the fact that, historically, the first such properties emerged in the study of exact sequences in the sense of Homological Algebra. The matrix properties  in the title of this post are particular types of exactness properties, which can be encoded using integer matrices. Before explaining what they are, let us first recall the notions of limit and colimit. Given a diagram of objects and arrows (objects are certain mathematical structures and arrows are morphisms between them), a limit (of the diagram) is a way to encode the information about the diagram in a single object; it is a terminal (...

The video shows two dancers, a man and a woman. The woman exists in the imagination of the man. Perhaps it is someone he knew, loved, cared for... Ballet-style gentle dance of the woman is contrasted with a free-style rough dance of the man. This symbolizes the unattainability of the imagination. Half-way in the video the man is shown collapsed on the floor. This scene is intended to reveal that the man has actually been suffering from the unattainability of his imagination. He longs for the woman, who remains to be only a fragment of his imagination throughout the video. Towards the end of the video, the woman dances to a light and playful music. The video ends with the man walking away from the camera. He momentarily stops, looks back, and then confidently makes a few more steps in the direction he was heading, away from the camera, till the video concludes. The man has resolved the problem: the joy of her existence, even if only in his imagination, must overcome the suffering.